Visualizing the charge order and topological defects in an overdoped (Bi,Pb)$_2$Sr$_2$CuO$_{6+x}$ superconductor

TL;DR

This study uses scanning tunneling microscopy to visualize charge order and topological defects in an overdoped cuprate superconductor, revealing their energy-dependent evolution and spatial dynamics.

Contribution

It provides the first detailed visualization of topological defects in charge order in overdoped cuprates, linking defect behavior to energy and pseudogap phenomena.

Findings

Charge order wavevector is non-dispersive with energy.

Topological defects decrease in density and migrate with increasing energy.

Pairs of defects can appear and disappear as energy varies.

Abstract

Electronic charge order is a symmetry breaking state in high-$T_\mathrm{c}$ cuprate superconductors. In scanning tunneling microscopy, the detected charge-order-induced modulation is an electronic response of the charge order. For an overdoped (Bi,Pb)$_2$Sr$_2$CuO$_{6+x}$ sample, we apply scanning tunneling microscopy to explore local properties of the charge order. The ordering wavevector is non-dispersive with energy, which can be confirmed and determined. By extracting its order-parameter field, we identify dislocations in the stripe structure of the electronic modulation, which correspond to topological defects with an integer winding number of $\pm 1$. Through differential conductance maps over a series of reduced energies, the development of different response of the charge order is observed and a spatial evolution of topological defects is detected. The intensity of charge-order-induced modulation increases with energy and reaches its maximum when approaching the pseudogap energy. In this evolution, the topological defects decrease in density and migrate in space. Furthermore, we observe appearance and disappearance of closely spaced pairs of defects as energy changes. Our experimental results could inspire further studies of the charge order in both high-$T_\mathrm{c}$ cuprate superconductors and other charge density wave materials.

Equations

$${\tilde{Z}}_f^x(\mathbf{q},\epsilon )=\tilde{Z}(\mathbf{q},\epsilon )[\tilde{f}(\mathbf{q}+{\mathbf{q}}_x^{\ast })+\tilde{f}(\mathbf{q}-{\mathbf{q}}_x^{\ast })],$$

$$Z_f^x(\mathbf{r},\epsilon )=A_x(\mathbf{r},\epsilon )\cos [{\mathbf{q}}_x^{\ast }\cdot \mathbf{r}+{\varphi }_x(\mathbf{r},\epsilon )],$$

$$\tan {\varphi }_x(\mathbf{r},\epsilon )=\frac{\int d{\mathbf{r}}^{\prime }{Z}_f^x({\mathbf{r}}^{\prime },\epsilon )\sin [{\mathbf{q}}_x^{\ast }\cdot {\mathbf{r}}^{\prime }]f(\mathbf{r}-{\mathbf{r}}^{\prime })}{\int d{\mathbf{r}}^{\prime }{Z}_f^x({\mathbf{r}}^{\prime },\epsilon )\cos [{\mathbf{q}}_x^{\ast }\cdot {\mathbf{r}}^{\prime }]f(\mathbf{r}-{\mathbf{r}}^{\prime })},$$

Full text

Introduction

The electronic charge order, accompanying the pseudogap (PG) state, is an interesting phenomenon in high-$T_{\mathrm{c}}$ cuprate superconductors keimer2015quantum ; timusk1999pseudogap ; Vojta2009stripe . In the electronic phase diagram, the PG state and the charge order emerge when a parent Mott insulator is doped with charge carriers cai2016visualizing . With the increase of doping level, superconductivity is developed and coexists with both the PG and charge order states over a broad doping range. The study of the PG and charge order states may help to unravel the mechanism of high-$T_{\mathrm{c}}$ superconductivity Vojta2009stripe ; cai2016visualizing ; griiner1994density ; hoffman2002four ; kohsaka2007intrinsic ; vershinin2004local ; parker2010silva ; da2014ubiquitous ; kohsaka2008cooper ; wise2008 ; wise2009 ; mesaros2011 ; fujita2014direct ; comin2014charge ; peng2017robust ; peng2016direct ; comin2015symmetry ; chang2012direct ; wu2011magnetic ; HamidianNatPhys2016 ; MesarosPNAS2016 ; fujita2014simultaneous ; lawler2010intra ; webbArxiv .

As a powerful tool of detecting the electronic structure with atomic resolution fischer , scanning tunneling microscopy (STM) has been extensively applied to investigate the charge order of cuprates  cai2016visualizing ; hoffman2002four ; kohsaka2007intrinsic ; vershinin2004local ; parker2010silva ; da2014ubiquitous ; kohsaka2008cooper ; wise2008 ; wise2009 ; mesaros2011 ; fujita2014direct ; fujita2014simultaneous ; lawler2010intra ; webbArxiv ; HamidianNatPhys2016 ; MesarosPNAS2016 . In an STM conductance map , the response of the charge order is usually identified by a checkerboard-like modulation along two perpendicular CuO bond directions. In the Fourier-transformed data, the checkerboard-like modulation is represented by four peaks centered at $\pm\bm{q}_{x}^{\ast}$ and $\pm\bm{q}_{y}^{\ast}$. The absolute values of the charge order peaks are inversely proportional to the periodicity $a_{0}/\delta$ of the real-space modulation, given by $|\bm{q}_{x}^{\ast}|=|\bm{q}_{y}^{\ast}|=2\pi\delta/{a_{0}}$ with $a_{0}$ the lattice constant. The checkerboard-like modulation with $\delta\approx 3/4$ has been well studied kohsaka2007intrinsic ; parker2010silva ; kohsaka2008cooper ; mesaros2011 ; fujita2014simultaneous ; lawler2010intra , and is prominent at high energies around the PG magnitude. Recently, the charge order of $\delta\approx 3/4$ has been discovered to be an electronic modulation with a $d$-form factor fujita2014direct ; HamidianNatPhys2016 ; MesarosPNAS2016 , compatible with the charge order of $\delta\approx 1/4$ detected with different experimental techniques Vojta2009stripe ; cai2016visualizing ; hoffman2002four ; vershinin2004local ; da2014ubiquitous ; wise2008 ; wise2009 ; comin2014charge ; peng2017robust ; peng2016direct ; comin2015symmetry ; webbArxiv .

The STM studies of the charge order are mainly implemented in two families of cuprates, Bi2Sr2CaCu2O8+x (Bi-2212) and Bi2Sr2CuO6+x (Bi-2201). With respect to the crystal structure, two CuO layers exist in a unit cell of Bi-2212, while only one in that of Bi-2201. The electronic structures of these two cuprates are different as well. The PG state vanishes in the middle of the superconducting regime in Bi-2212 fujita2014simultaneous , while it extends to the overdoped regime in Bi-2201 heyang2014 ; zheng2017the . Therefore the charge order in Bi-2201 can be explored in a broad range of doping. The determination of the ordering wavevector has been a main focus in previous studies MesarosPNAS2016 ; webbArxiv . With the increase of doping, a commensurate to incommensurate transition of the wavevector has been discovered webbArxiv .

The structural disorder in doped cuprates can induce spatial fluctuations in the electronic orders. In the Ginzburg-Landau theory, the charge order can be described by an order-parameter field $\psi(\bm{r})=A(\bm{r})\exp[i\phi(\bm{r})]$, where the amplitude $A(\bm{r})$ and the phase $\phi(\bm{r})$ vary spatially. In general, charge order is a long-range order with local fluctuations resulting from perturbations. Furthermore, dislocations may interrupt the long range order, with dislocation cores represent singularities in the order parameter phase $\phi(\bm{r})$. A nonzero integer winding number is obtained along any path enclosing such a singular point, thus named a topological defect. Previous work examining underdoped Bi-2212 extracted the spatial structure of local fluctuations in the charge-order-induced modulation, revealing a close connection to nematicity mesaros2011 . A similar study examining the local structure of the charge-order-induced signal in Bi-2201 is a natural extension.

In this paper, we apply STM to study the charge order in overdoped Bi-2201. A non-dispersive and incommensurate wavevector is determined to be at $\delta\approx 0.83$. From the local amplitude and phase of the modulation, we identify singular topological defects in the order parameter field, corresponding to dislocation cores in the filtered stripe structure of the charge-order-induced modulation. Around a single defect, the data analysis leads to a generic phase slip picture. As energy increases away from the Fermi level, the response of the charge order is gradually enhanced, and the topological defects migrate in space. The appearance and disappearance of defect pairs are also observed. The defect ‘movement’ can be related to the transfer of dislocation cores in regions of bent stripes. The total number of defects decreases with energy until it saturates, which implies that the intensity of modulation can affect how robust the response of charge order is against local perturbations. This energy-dependent phenomenon can be further investigated in other cuprate superconductors and charge density wave materials.

Materials and method

The high-quality (Bi,Pb)2Sr2CuO6+x single crystal in our experiment is grown by the traveling-solvent floating-zone method lin2010high . With a size of $2$ mm $\times$ 2 mm $\times$ 0.1 mm, the sample is cut from an as-grown ingot, followed by being post-annealed at a specific temperature and under a controlled atmosphere. The critical temperature of the Pb-doped Bi-2201 sample studied in this work is $T_{c}\approx 13$ K. The hole doping for this overdoped (OD13K) sample is estimated to be $p\approx 0.21$ according to the Ando formula andoformula ; FeiyingSCPMA .

All the data in this paper are taken in an ultra-high-vacuum STM zheng2017the . The Bi-2201 sample is cleaved at liquid-nitrogen temperature and immediately inserted into the STM head. The measurement is performed at liquid-helium temperature ($T\approx 4.5$ K). The STM topography is taken at a sample bias of $V_{\mathrm{b}}=100$ mV and a setpoint current of $I_{\mathrm{s}}=100$ pA. The local differential conductance ($dI/dV$) spectra are acquired simultaneously with the topography by a standard lock-in technique with a modulation frequency of $f=983.4$ Hz.

Results

Topography and charge order of overdoped Bi-2201. We work with a high-quality overdoped (Bi,Pb)2Sr2CuO6+x single crystal sample lin2010high ; zheng2017the whose critical temperature $T_{c}\approx 13$ K and hole doping $p\approx 0.21$ andoformula ; FeiyingSCPMA . The data is taken in an ultra-high-vacuum STM at $T\approx 4.5$ K zheng2017the . Figure 1(a) is a topographic image of 27 nm $\times$ 27 nm taken on a cleaved BiO surface, showing a square lattice of Bi atoms with the lattice constant $a_{0}\approx 3.8$ Å. In the CuO layer, each Cu atom is located directly below a Bi atom, sharing the same lattice constant HamidianNJP2012 . The brighter spots in Fig. 1(a) are Pb atoms, which are partially substituted for Bi atoms and strongly suppress the structural supermodulation.

The differential conductance spectra are measured by varying the voltage $V$, in which the PG magnitudes ($\Delta_{\mathrm{PG}}$) can be extracted from coherence-peak positions. The local electronic structure is probed by a differential conductance map, $g(\bm{r},E=eV)=dI/dV(\bm{r},V)$. In our Pb-doped Bi-2201 sample, the distribution of both the differential conductance and the PG magnitude is spatially inhomogeneous (see Supplementary I). To exclude the spatial inhomogeneity of the PG magnitude, the conductance map is rescaled as $g(\bm{r},\varepsilon)$ with a reduced energy $\varepsilon=E/\Delta_{\mathrm{PG}}(\bm{r})$. Subsequently, we calculate a ratio map, $Z(\bm{r},\varepsilon)=g(\bm{r},\varepsilon)/g(\bm{r},-\varepsilon)$, which reduces the systematic error known as the setpoint effect in STM measurement kohsaka2007intrinsic ; kohsaka2008cooper ; heyang2014 . In this field of view (FOV), we also find some zero gap patches which can be attributed to the van Hove singularity (VHS) states fischer ; zheng2017the ; yayuNJP . A spatially averaged value of $Z(\bm{r},\varepsilon)$ is applied to fill in the VHS regions zheng2017the . As the differential conductance is proportional to the electronic density of states, the response of charge order is reflected in the $Z(\bm{r},\varepsilon)$-maps at different reduced energies, which is the main focus of data analysis in this paper.

Figure 1(b) displays a ratio map, $Z(\bm{r},\varepsilon=1.05)$, at the energy scale of the PG and in the same FOV as in Fig. 1(a). A small deviation of $\varepsilon$ from unity is due to our technique of binning the reduced energies. A checkerboard-like spatial modulation is observed, which represents a response of the static charge order in recorded STM results. To maintain an atomic registry across the FOV, drift correction of the lattice is applied to both topographic and electronic data lawler2010intra . To quantify the spatial periodicity of charge-order-induced modulation at the PG energy, the Fourier transform of Fig. 1(b), $\tilde{Z}(\bm{q},\varepsilon)=\mathrm{FT}[Z(\bm{r},\varepsilon)]$, is shown in the momentum $\bm{q}$-space in Fig. 1(c). Four sharp Bragg peaks located at $\pm\bm{Q}_{x}$ and $\pm\bm{Q}_{y}$ are observed, each collapsed into a single pixel due to the lattice drift correction zheng2017the ; lawler2010intra . The Bragg wavevectors, $\bm{Q}_{x}=(2\pi/{a_{0}},0)$ and $\bm{Q}_{y}=(0,2\pi/{a_{0}})$, are consistent with the square lattice structure of Bi atoms. In addition, there exist four peaks near the Bragg peaks, which correspond to the checkerboard-like spatial modulation induced by the charge order. The centers of the four charge order peaks are estimated to be at $\pm\bm{q}^{\ast}_{x}\approx\pm 0.83\bm{Q}_{x}$ and $\pm\bm{q}^{\ast}_{y}\approx\pm 0.83\bm{Q}_{y}$ from Gaussian fitting (Supplementary II). The same analysis can be applied to $\tilde{Z}(\bm{q},\varepsilon)$-maps at different reduced energies. In the top panel of Fig. 1(d), we plot the amplitude $|\bm{q}^{\ast}_{x/y}|$ of the charge-order wavevectors as a function of reduced energy $\varepsilon$. As $\varepsilon$ changes, the wavevectors $\bm{q}^{\ast}_{x/y}$ are nearly invariant, suggesting a non-dispersive and static charge order. The wavevectors are incommensurate with the lattice, consistent with the result for overdoped Bi-2201 in Ref. webbArxiv . In the bottom panel of Fig. 1(d), we also plot the intensity of the charge-order peak as a function of $\varepsilon$, showing that the response of the charge order is weak at small values of $\varepsilon$ and becomes intensified around the PG energy. A more detailed discussion of the $\varepsilon$-dependence will be provided later.

Identification of topologaical defects. To explore the spatial variation of the checkerboard-like modulation, we first separate the $x$- and $y$-components with a Gaussian filtering technique. The $x$-component of the modulation in the momentum space is extracted as

[TABLE]

where $\tilde{f}(\bm{q})=\exp(-q^{2}\Lambda^{2}/2)$ is a Gaussian filtering function. In practice, the cutoff size is chosen at $\Lambda=1.2$ nm. Following an inverse Fourier transform, $Z^{x}_{f}(\bm{r},\varepsilon)=\mathrm{FT}^{-1}[\tilde{Z}^{x}_{f}(\bm{q},\varepsilon)]$, we obtain the checkerboard-like modulation along the $x$-direction in the real space. The same technique is applied to extract the $y$-components, $\tilde{Z}^{y}_{f}(\bm{q},\varepsilon)$ and $Z^{y}_{f}(\bm{r},\varepsilon)$. The final results of $Z^{x}_{f}(\bm{r},\varepsilon=1.05)$ and $Z^{y}_{f}(\bm{r},\varepsilon=1.05)$ at the PG energy are plotted in Figs. 2(a) and 2(d), from which we observe stripes along the $y$- and $x$-directions, respectively. In general, the $x$-component of the charge order can be re-expressed as

[TABLE]

where $A_{x}(\bm{r},\varepsilon)$ and $\phi_{x}(\bm{r},\varepsilon)$ are the amplitude and phase of the order-parameter field at location $\bm{r}$. Here another Gaussian filtering process is applied to extract $\phi_{x}(\bm{r},\varepsilon)$ around the charge order wavevector mesaros2011 , given by

[TABLE]

with $f(\bm{r})=\mathrm{FT}^{-1}[\tilde{f}(\bm{q})]$. The amplitude $A_{x}(\bm{r},\varepsilon)$ is then calculated using the definition in Eq. (2). Accordingly, the order-parameter field associated with the $x$-component of the charge order is given by $\psi_{x}(\bm{r})=A_{x}(\bm{r})\exp[i\phi_{x}(\bm{r})]$. In Figs. 2(b) and 2(c), we show $A_{x}(\bm{r},\varepsilon=1.05)$ and $\phi_{x}(\bm{r},\varepsilon=1.05)$ extracted from Fig. 2(a). The amplitude and phase are inhomogeneous, unlike the uniform order parameter of ideal long range order. The three spatially resolved quantities, $Z^{x}_{f}(\bm{r},\varepsilon=1.05)$, $A_{x}(\bm{r},\varepsilon=1.05)$ and $\phi_{x}(\bm{r},\varepsilon=1.05)$ in Figs. 2(a)-2(c), display correlated behavior. The same approach is applied to the $y$-component of the charge-order-induced modulation ($Z^{y}_{f}(\bm{r},\varepsilon=1.05)$ in Fig. 2(d)), and the resulting $A_{y}(\bm{r},\varepsilon=1.05)$ and $\phi_{y}(\bm{r},\varepsilon=1.05)$ are plotted in Figs. 2(e) and 2(f), respectively.

From Figs. 2(a) and 2(d), we observe that the stripes of charge-order-induced modulation are distorted in space, including weak fluctuations, as well as strong disruptions. As an illustration, we select a small area in the solid box in Fig. 2(a) and plot magnified maps of $Z^{x}_{f}(\bm{r},\varepsilon=1.05)$, $A_{x}(\bm{r},\varepsilon=1.05)$ and $\phi_{x}(\bm{r},\varepsilon=1.05)$ in Figs. 3(a)-3(c). Near the middle of the image, a single stripe splits into two stripes along the $y$-direction, analogous to an edge dislocation in a crystal lattice. The manifestation of the branch point, or dislocation core, in $A_{x}(\bm{r})$ and $\phi_{x}(\bm{r})$ is associated with the presence of a singularity: (1) The phase $\phi_{x}(\bm{r})$ is undefined at the dislocation core, while a winding phase of $2\pi$ is acquired for $\phi_{x}(\bm{r})$ if a clockwise cycle is taken around this singularity. (2) To generate a physically meaningful order parameter, the corresponding amplitude $A_{x}(\bm{r})$ is suppressed to zero, as shown by a dark region near the middle of Fig. 3(b). To further visualize the behavior of the singularity, seven linecuts along the $y$-direction are chosen around the dislocation in Fig. 3(a). For each linecut, the real and imaginary parts of $\psi_{x}(\bm{r})\exp[i\bm{q}_{x}^{\ast}\cdot\bm{r}]$ are plotted as a function of the spatial location $\bm{r}$ (see Fig. 3(d)). Along the right top linecut (in blue color), a spiral of $\psi_{x}(\bm{r})\exp[i\bm{q}_{x}^{\ast}\cdot\bm{r}]$ propagates with a roughly constant amplitude and four turns are generated within the selected distance. As the linecut approaches the dislocation core (the green line), the spiral is disrupted and the middle turn is broken into two very small turns. An extra phase of $2\pi$ is acquired through $\phi_{x}(\bm{r})$ so that the number of the spiral turns is increased from 4 to 5 along the linecuts below the green one, similar to the phase slip picture for other ordered states chaikin1995principles ; tinkham1996introduction . Through a combined analysis of $Z^{x/y}_{f}(\bm{r},\varepsilon)$, $A_{x/y}(\bm{r},\varepsilon)$ and $\phi_{x/y}(\bm{r},\varepsilon)$, the dislocation cores behave as topological defects in the charge-order-induced modulation, and almost do not affect the wavevector of the modulation (see Fig. S3 in Supplementary III).

With $\phi_{x/y}(\bm{r})$ defined modulo $2\pi$, a branch cut following a curved line can be emanating from each defect (see Figs. 2(c) and 2(f)). Crossing a branch cut generates an artificial jump of $\pm 2\pi$, which however does not affect the value of the order parameter $\psi_{x/y}(\bm{r})$. In this FOV, most of these branch cuts connect two defects with opposite polarities defined by the signs of their winding phases. In general, topological defects are emergent excitations in an ordered state, such as quantized vortices in superfluids and superconductors chaikin1995principles ; tinkham1996introduction . Compared to a single unbounded defect, a pair of defects with opposite polarities have a lower excitation energy associated with an attractive interaction. In our system, defect pairing is observed in Figs. 2(c) and 2(f), except for those around the corners of the FOV.

Evolution of the charge-order-induced modulation and topological defects. In cuprates such as Bi-2201 superconductors, various electronic orders coexist and can compete with each other. In Fig. 1(d), we find that response of the charge order appears at a low $\varepsilon$ and is intensified at the PG energy, consistent with a consensus that the charge order is correlated with the PG state parker2010silva ; fujita2014simultaneous ; vershinin2004local ; wise2009 ; comin2014charge ; heyang2014 . Next we explore the spatial variation of the charge-order-induced modulation at different reduced energies. The above analysis procedure is applied to $Z(\bm{r},\varepsilon)$ maps of different $\varepsilon$ to obtain the $x$- and $y$-components of the modulation, as well as their amplitudes and phases. From a series of the filtered maps of $Z^{x}_{f}(\bm{r},\varepsilon)$ and $Z^{y}_{f}(\bm{r},\varepsilon)$ (see Supplementary Fig. S6 and S8), we observe that the stripe structure of the modulation is maintained when $\varepsilon$ increases. However, we also find visible changes at certain regions. The one-dimensional structure of a stripe is easily bent by local stress. Neighboring stripes are bent similarly but with gradually decaying strengths. Under large stress, dislocations can be created among the bent stripes. As $\varepsilon$ changes, this local stress may be released or enhanced, causing ‘movement’ of defects and even their appearance and disappearance. We emphasize that the charge order itself is a static electronic order without energy dependence. The $\varepsilon$-dependent phenomenon discussed here is the different response of the charge order represented in the STM spectroscopy measurement.

To visualize the $\varepsilon$-dependence of the detailed stripe structure, we inspect the phase maps of $\phi_{x/y}(\bm{r},\varepsilon)$. As shown in Figs. 4(a) and 4(b), the overall structure of two phase maps, $\phi_{x}(\bm{r},\varepsilon=0.45)$ and $\phi_{x}(\bm{r},\varepsilon=1.05)$, are similar to each other, demonstrating the stability of the charge-order-induced modulation but with subtle changes. As shown in Fig. 4(c), the total number of defects, extracted from both $\phi_{x}(\bm{r},\varepsilon)$ and $\phi_{y}(\bm{r},\varepsilon)$, gradually decreases with $\varepsilon$ and is stabilized around 30 when $\varepsilon$ is larger than 0.5. We record the spatial locations of defects at different $\varepsilon$ and present a three-dimensional map in Fig. 4(d), showing the movement of defects. The appearance and disappearance of defect pairs are observed, and can even occur around the PG energy (see Supplementary VI for more details). Two examples are shown in Fig. 4(e). In the left column, the distance between two defects gradually decreases with the increasing $\varepsilon$. For $\varepsilon=0.9$ and 1.05, the two defects cannot be distinguished from each other because their distance is smaller than our cutoff size, demonstrating ‘annihilation’ of a defect pair with opposite polarities. The right column shows the ‘creation’ of a defect pair connected by a branch cut. For charge-order-induced modulation with a relatively low intensity, the stripe bending is prone to induce pairs of closely spaced defects, leading to the increasing number of defects at small $\varepsilon$. The ‘movement’ of defects corresponds to transfer of dislocation cores within a region of bent stripes (see Supplementary Fig. S10). Furthermore, we find a negligible correlation between the topological defects and inhomogeneous PG (see Supplementary IV). Although the spatial resolution of detecting paired defects changes with the cutoff size of the filtering function (or the coarse grained length equivalently), the evolution of defects qualitatively holds for various cutoff sizes (see Supplementary V).

Conclusions

In this paper, we investigate the charge order in an overdoped (Bi,Pb)2Sr2CuO6+x (Bi-2201) sample with STM. In real space, the charge-order-induced modulation is identified in the ratio $Z$-map of $Z(\bm{r},\varepsilon)$. In momentum space, the modulation is represented by four peaks around incommensurate wavevectors $\pm\bm{q}^{\ast}_{x}\approx\pm 0.83(2\pi/{a_{0}},0)$ and $\pm\bm{q}^{\ast}_{y}\approx\pm 0.83(0,2\pi/{a_{0}})$. The charge-order-induced modulation starts to appear at low reduced energies. With increasing $\varepsilon$, the modulation is gradually intensified and reaches its maximum strength around the PG energy. The incommensurate wavevector is not dispersive with $\varepsilon$, consistent with the static charge order or charge density wave order. The spatially-varied order-parameter fields enable identification of singular points, i.e. topological defects. In consecutive order-parameter maps with changing $\varepsilon$, the positions of defects gradually change, and pairs of defects appear and disappear. As energy approaches the PG energy, the number of defects decreases and saturates. This phenomenon uncovers a new aspect of the charge-order response in STM measurement. We expect to investigate whether the observed behavior is generic in cuprate superconductors or other charge density wave materials, which requires future experimental and theoretical efforts.

Acknowledgements

This work is supported by the National Basic Research Program of China (2015CB921004), the National Natural Science Foundation of China (NSFC-11374260), and the Fundamental Research Funds for the Central Universities in China. XJZ thanks financial support from the National Natural Science Foundation of China (NSFC-11334010), the National Key Research and Development Program of China (2016YFA0300300), and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (XDB07020300).